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April 17, 2013

Koudenburg on Algebraic Weighted Colimits

Posted by Simon Willerton

My student Roald Koudenburg recently successfully defended his thesis and has yesterday put his thesis on the arXiv.

Roald Koudenburg, Algebraic weighted colimits

I will give a rough caricature of what he does. For a much nicer overview, I suggest you read the well-written introduction to the thesis! (Relating to the Café, there’s even an example including Simon Wadsley’s Theorem into Coffee.)

Roald was originally thinking about Ezra Getzler’s approach to operads in Operads revisited, and needed to generalize a result of Getzler’s on when the left Kan extension of a symmetric monoidal functor along a symmetric monoidal functor is itself a symmetric monoidal functor.

If you are of that kind of persuasion, you will be aware that symmetric monoidal categories are the pseudo-algebras for the free symmetric strict monoidal category 22-monad on Cat{Cat}, the 22-category of categories, and that symmetric monoidal functors are the algebra maps for this 22-monad.

You might then consider the situation of a 22-monad TT on a 22-category, and ask when the left Kan extension of a map of TT-algebras along a map of TT-algebras is again a TT-algebra map.

Here are three examples of algebras for a 22-monad that Roald considers.

  1. Ordered compact Hausdorff spaces are algebras for the ultrafilter 22-monad on 22-Cat. In this case the question becomes, when is the left Kan extension of a continuous order preserving map along another such map also continuous and order preserving?

  2. Double categories can be considered as algebras for a certain 22-monad on a 22-category of internal categories in a specific presheaf category. (Yes, I find that a mouthful.)

  3. Similarly, monoidal globular categories of Batanin are algebras for some 22-monad.

Going back to the symmetric strict monoidal category monad, this monad actually extends from the 22-category CatCat to the the proarrow equipment of categories, functors and profunctors. Mike wrote a nice post here at the Café on Equipments and what they have to do with limits.

It turns out that there are several examples of 22-monads on equipments, with interesting algebras and Roald looks at conditions necessary for when, given such a 22-monad TT, the left Kan extension of a map of TT-algebras along map of TT-algebras is a map of TT-algebras.

I should at this point say something precise about the main result: given a ‘right suitable normal’ monad TT on a closed equipment KK, Roald defines ‘right colax TT-promorphisms’ which, together with the usual colax T-algebra maps, form an equipment T-rcPromT\text{-rcProm}. The main theorem is the following.

Theorem: Let TT be a ‘right suitable normal’ monad on a closed equipment KK. The forgetful functor UT:T-rcPromKUT: T\text{-rcProm} \to K ‘lifts’ all weighted colimits. Moreover its lift of a weighted colimit colim Jd:BMcolim_J d: B\to M, where d:AMd: A \to M is a pseudomorphism and J:ABJ : A \to B is a right pseudopromorphism, is a pseudomorphism whenever the canonical vertical cell colim T(mTd)mT(colim Jd) colim_{T} (m \circ Td) \Rightarrow m\circ T(colim_J d) is invertible, where m:TMMm: TM \to M is the structure map of MM.

Posted at April 17, 2013 6:00 PM UTC

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Re: Koudenburg on Algebraic Weighted Colimits

Congratulations, both!

Posted by: Tom Leinster on April 18, 2013 12:24 AM | Permalink | Reply to this

Re: Koudenburg on Algebraic Weighted Colimits

Visitors to the University of Sheffield. may have heard tell of the million-pound full stop, which — as the result of an expensive rebranding exercise — must now appear at the end of the phrase “University of Sheffield.”, and names of departments, on all letterheads, websites, etc. You can see it on signs as you walk around town.

With that in mind, I was amused to see Roald’s title page, where he (rather unusually) puts a full stop after his name, and also after “A thesis submitted for the degree of Doctor of Philosophy”, and then after the date submitted… but not after “The University of Sheffield”

Is this an in-joke?

Posted by: Tom Leinster on April 18, 2013 12:35 AM | Permalink | Reply to this

Re: Koudenburg on Algebraic Weighted Colimits

Thanks Tom!

I’m afraid not much thought has gone into the full stops. When typing up the title page I looked at someone else’s which also had them. I remember thinking shortly about whether to add them before doing so.

But you are right about it being unusual: I just looked at some other theses; none of them had full stops in their title pages.

I’m happy the university of Sheffield doesn’t require a lay summary!

Posted by: Roald Koudenburg on April 22, 2013 8:40 PM | Permalink | Reply to this

Re: Koudenburg on Algebraic Weighted Colimits

I’ve just discovered that PhD students at Edinburgh have to submit a “lay summary” with their thesis. From the guidance notes:

The lay summary must be no longer than can be accommodated in single space type on one side only of a single form …

A lay summary is intended to facilitate knowledge exchange, public awareness and outreach. It should be in simple, non-technical terms that are easily understandable by a lay audience, who may be non-professional, non-scientific and outside the research area.

Abstracts, particularly in science, engineering, medicine and veterinary medicine, may be highly technical or contain scientific language that is not easily understandable to readers outside the research area. Therefore, the lay summary is intended as supplementary to the abstract.

What would Roald’s have said?

Posted by: Tom Leinster on April 18, 2013 5:57 PM | Permalink | Reply to this

Re: Koudenburg on Algebraic Weighted Colimits

Indeed, congratulations Roald!

and Roald looks at conditions necessary for when, given such a 2-monad T, the left Kan extension of a map of T-algebras along map of T-algebras is a map of T-algebras.

Does the main theorem mentioned above talk to these conditions?

Posted by: Bruce Bartlett on April 19, 2013 12:04 PM | Permalink | Reply to this

Re: Koudenburg on Algebraic Weighted Colimits

Oh, sorry I see from the abstract that a left Kan extension is to be thought of as an example of a weighted colimit.

Posted by: Bruce Bartlett on April 19, 2013 12:08 PM | Permalink | Reply to this

Re: Koudenburg on Algebraic Weighted Colimits

Thanks Bruce!

Hoe gaat het? Simon vertelde me dat je in Sheffield bent nu, vind je het leuk er weer eens te zijn? (en koud ook zeker? ;))

Posted by: Roald Koudenburg on April 22, 2013 8:45 PM | Permalink | Reply to this

Re: Koudenburg on Algebraic Weighted Colimits

Dankie, gaat goed hier, terug waar mense jou “luv” noem. Nie echt koud nie, en die snooker is terug, wat leuk is! Verskoon my aaklike mengsel van het nederlands en afrikaans. Ek stuur vir jou my beste wense vir die toekoms.

Posted by: Bruce Bartlett on April 24, 2013 12:34 PM | Permalink | Reply to this

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